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I am hoping someone can help me clear up my understanding on rotational dynamics. In particular, how to determine the torque required to keep an object rotating.

My understanding (please tell me if it is wrong):

  1. An object will continue rotating at constant angular velocity L as long as the axis of rotation is an axis of symmetry.

  2. If an object is not rotating about an axis of symmetry then the object will slow down because the vector L is constantly changing and is not parallel to the axis.

This means that for such an unbalanced rotation, a torque would need to be applied about the axis to maintain the angular velocity.

Consider a rod of length L that has uniform mass and is attached to a fixed axis at one of its ends. If this object is given an angular velocity $\omega$ then I believe that this would need to be maintained by supplying a constant torque about the axis, since this object is not rotating about an axis of symmetry.

However, I don't understand how to demonstrate why this is so. That is, how to calculate the angular momentum vector. My understanding that this is the sum of the angular momentums of all the particles, or the sum of $r \times p$ for each particle. But this is where I can't follow what my textbook is saying. Is r measured from the centre of mass or from the axis? No matter how I try and work it out it seems the net angular momentum works out to be parallel to the axis and in the same direction as the angular velocity vector. But that contradicts my understanding!

As a side question, can anyone point me to a good online tutorial on rotational dynamics. My current textbook has been great at explaining everything up to this point, but I find the rotational dynamics section confusing.

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  • $\begingroup$ As long there is no friction, the rotation will not slow down, regardless of the axis. $\endgroup$ – Norbert Schuch Jan 11 '16 at 10:40
  • $\begingroup$ If this is true I'm totally confused. One of the problems in my textbook specifically asks for the required torque required to keep an unbalanced object rotating at constant angular velocity. There is no mention of friction, so I assume it's a frictionless bearing. $\endgroup$ – esotechnica Jan 12 '16 at 0:35
  • $\begingroup$ Well, an unbalanced object will possibly not rotate nicely (e.g., precess), in which case one might try to hold it in balance by applying a torque without doing any work. $\endgroup$ – Norbert Schuch Jan 12 '16 at 0:40
  • $\begingroup$ I don't have my book with me at the moment but I will have a look again tonight. I'm wondering if it is asking for the torque that the axle must resist. That is, the component of the angular momentum that is perpendicular to the axis. I think this is what your saying? $\endgroup$ – esotechnica Jan 12 '16 at 0:50
  • $\begingroup$ Precisely. Imagine the axis is mounted to an infinitely heavy block. Then the object will nicely rotate, but the axis/block will experience some torque. $\endgroup$ – Norbert Schuch Jan 12 '16 at 1:03

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